Using Parallax to Measure Distance

Last time we introduced an optical effect known as the principle of parallax. Today we’ll get a step closer to seeing how it’s instrumental in measuring distances. For our example we’ll be viewing a point on a tree several blocks away.

Since we can’t measure the distance, r, to the tree directly by using a tape measure, we’ll have to use another approach that is made available to us through the principle of parallax. The first step to doing this is to establish two different lines of sight to the red dot on the tree. For our example these will originate at Points A and B, which are both viewing the same spot.

Line of Sight A extends straight from Point A to the red spot, while Point B, which is situated only a few feet to the right of A, will provide us with a slanted line of sight to it. We’ll refer to the path between Points A and B as Path AB, which is represented by a black dashed line and has a length, d.

You may have noticed from the illustration that Line of Sight B, because it lies on a slant relative to Line of Sight A, provides us with an internal angle, θ, relative to Path AB. This relationship is in fact required to exist if we are to use the principle of parallax, and you’ll see why later.

Another requirement is that Path AB must be at a right angle to Line of Sight A, which it is. The illustration bears this out by the fact that a right triangle is formed when Lines of Sight A, B, and Path AB intersect.

Right triangles are generally regarded as special within mathematics, but they are particularly special when applying the principle of parallax. The presence of a right triangle enables us to use a branch of mathematics known as trigonometry to correlate the triangle’s angles to the ratios of the lengths of its sides. In other words, we can calculate r by using trigonometry if we are able to measure d and θ.